Pricing and risk measures of mortgage backed securities with PDE method

Transcription

1 Pricing and risk measures of mortgage backed securities with Xuefei He Manchester Business School October, /14

2 Content Mortgage backed securities Mortgage backed securities 2/14

3 What is mortgage backed security (MBS)? House buyer Buy house with loans Pay monthly installment Prepayment and default options Fannie Mae, Freddie Mac, Ginnie Mae and banks Issue MBS based on the loans mortgage pool Pass the monthly payments to MBS investors Earn commission fee Remove those loans from their balance sheet MBS investors Get payment and bear risk Higher return than other fixed income products Almost no default risk of the three agencies 3/14

4 Why causing crisis Bubble: MBS investors invest heavily in MBS Fannie Mae, Freddie Mac, Ginnie Mae and banks issue more MBS by attracting more house buyers House buyer speculators and subprime borrowers Rising house price and booming economy Burst: Lower house price House buyers default their loans Fannie Mae, Freddie Mac, Ginnie Mae and banks bailed out MBS investors bankrupt Magnified and spread by complex derivatives Crisis! 4/14

5 Modeling difficulties path dependent Normal fixed income product Interest rate path independent Price is the discounted expected future cash flow. MBS path dependent Prepayment both pass-through and CMO (Collateralized Mortgage Obligation) Complex tranche allocation CMO 5/14

6 Cash flows mathematical presentation of the figure Pass-through: where CF m = PR m +I m +PP m = MP m +PP m PP m = (B m 1 PR m ) SMM m, Sequential-pay CMO: m m m > m Tranche I PR m +PP m +Bm I r c 0 Tranche II B0 II PR m +PP m +Bm II 6/14

7 Steps: Mortgage backed securities Generate short rate path (r 1, r 2,, r 360 ), for example, Hull-White model; Calculate discounting factor DC m at month m, m = 1,2,,360 following DC m = m j=0 e (rm) ; Compute cash flows CF m ; Then the present value of all the cash flows along this short rate path i is equal to PV i = 360 t=1 CF m DC m ; Above steps are repeated N times. The MBS price is calculated as the average of all the PV i. 7/14

8 Time consuming for VaR calculation VaR is the maximum loss value in next N(=10) days with X%(=99%) confidence level. Loss distribution is calculated from historical interest rate data. One is in another simulation. 8/14

9 Equations Mortgage backed securities One-factor Hull-White model dr = (θ(t) ar)dt +σdz PDE for MBS P t + 1 P 2 σ2 2 r 2 +(θ(t) ar) P r + da dt P A rp = 0 9/14

10 Solution techniques Technique Advantage da/dt = 0 except 2D PDE > some discrete time points series of 1D PDE (month ends) Good choice of A, simpler operation A m = B m Bm SMM=0 A 0 = 1 Implicit finite difference at month ends More stable 10/14

11 Pass-through results Table: PDE results for pass-through. The original mortgage balance B 0 = 100 units, the pass-through rate r c = 5.5% and loan term n = 360 months. N r is the number of grid points in the r direction and N A is the number of grid points in the A direction. PDE Monte Carlo N r N A Price Time (s) N Price Time (s) /14

12 CMO results Mortgage backed securities Table: PDE results for CMO (tranche I). The original mortgage balance B 0 = 100 units, the pass-through rate r c = 5.5% and loan term n = 360 months. N r is the number of grid points in the r direction and N A is the number of grid points in the A direction. PDE Monte Carlo N r N A Price Time (s) N Price Time (s) /14

13 VaR calculation Calculate historical 10-day yield curve changes (δs = s i+2 s i ) and swaption volatility changes (δσ swaption market = σ swaption market,i+2 σswaption market,i ); Add the δs and δσ swaption market on today s yield curve (s 0 ) and swaption volatility (σ swaption market,0 ) to have s i and σ swaption market,i ; Compute the prices P i based on s i and σ swaption market,i with our PDE method; Obtain loss distribution L i = P i P 0 ; and VaR is the first percentile of L i. 13/14

14 VaR results Mortgage backed securities Weekly yield curves and swaption volatilities from Oct 2002 to Jun 2010, 403 days totally About 20 seconds to calculate VaR for one security VaR results from (13 11 grid points) VaR Pass-through 2.16 CMO tranche I 0.59 CMO tranche II /14

15 and highlight Industry standard Time consuming One MBS price One VaR One portfolio with one thousand MBS 1 second 10 minutes 1 week Our new Fast reduce 1 week to a few hours Flexible valid for most MBS products 15/14